Boundary Value Problems
Volume 2006 (2006), Article ID 75458, 12 pages
doi:10.1155/BVP/2006/75458
On explicit and numerical solvability of parabolic initial-boundary
value problems
1Department of Mathematics, University of Haifa, Haifa 31905, Israel
2Department of Mathematics, Natural Sciences Programs, Lesley Collage, Lesley University, 29 Everett Street, Cambridge 39762, MA, USA
Received 26 June 2005; Revised 15 January 2006; Accepted 22 March 2006
Copyright © 2006 Alexander Kozhevnikov and Olga Lepsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A homogeneous boundary condition is constructed for the parabolic
equation (∂t+I−Δ)u=f in an arbitrary
cylindrical domain Ω×ℝ (Ω⊂ℝn being a bounded domain, I and Δ being the
identity operator and the Laplacian) which generates an
initial-boundary value problem with an explicit formula
of the solution u. In the paper, the result is obtained not just
for the operator ∂t+I−Δ, but also for an
arbitrary parabolic differential operator ∂t+A, where
A is an elliptic operator in ℝn of an even order
with constant coefficients. As an application, the usual
Cauchy-Dirichlet boundary value problem for the homogeneous
equation (∂t+I−Δ)u=0 in Ω×ℝ is reduced to an integral equation in a thin lateral boundary
layer. An approximate solution to the integral equation generates
a rather simple numerical algorithm called boundary layer
element method which solves the 3D Cauchy-Dirichlet problem
(with three spatial variables).